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Category and Functor

Category defintions & Examples

Dec 15, 2015
 Extract from : CRing Project, Chapter 0 source : https://math.berkeley.edu/~amathew/chcategories.pdf

1. Category definition

In category theory, a branch of mathematics, the functors between two given categories form a category, where the objects are the functors and the morphisms are natural transformations between the functors. Functor categories are of interest for two main reasons: many commonly occurring categories are (disguised) functor categories, so any statement proved for general functor categories is widely applicable; every category embeds in a functor category (via the Yoneda embedding); the functor category often has nicer properties than the original category, allowing certain operations that were not available in the original setting.

2. Examples and notations

1.6 Preorder 2 Examples of categories 2.1 Baby examples 2.2 The category of sets 2.3 Mathematical structures as categories 2.3.1 Preorders 2.3.2 Groups 2.3.3 Matrices 2.4 Categories of sets with structure 3 Properties of objects and morphisms erminology and fine points If f: A to B in a category, A is called the domain or source of f, and B is called the codomain or target of f. text{Hom}(A,B) is called a hom class or a hom set if it is indeed a set. In general a hom set may be empty, but for any object A, text{Hom}(A,A) is not empty because it contains the identity morphism. The hom class text{Hom}(A,B) may be denoted by text{Hom}_mathcal{C}(A,B) or mathcal{C}(A,B) if it is necessary to specify which category is referred to. Morphisms may also be called maps. This does not mean that every morphism in any category is a set function (see #Baby examples and #Preorders). Arrow is a less misleading name. The composite gf may be written gcirc f. It might be more natural to write the composite of f:Ato B and g:Bto C as f g instead of g f but the usage given here is by far the most common. This stems from the fact that if the arrows are set functions and xin A, then (gf)(x)=g(f(x)). Thus g f is best read as "do g after f".

A category is a mathematical structure, like a group or a vector space, abstractly defined by axioms. Groups were defined in this way in order to study symmetries (of physical objects and equations, among other things). Vector spaces are an abstraction of vector calculus. What makes category theory different from the study of other structures is that in a sense the concept of category is an abstraction of a kind of mathematics. (This cannot be made into a precise mathematical definition!) This makes category theory unusually self-referential and capable of treating many of the same questions that mathematical logic treats. In particular, it provides a language that unifies many concepts in different parts of math. In more detail, a category has objects and morphisms or arrows. (It is best to think of the morphisms as arrows: the word “morphism” makes you think they are set maps, and they are not always set maps. The formal definition of category is given in the chapter on categories.)
Aspects of category theory Because the concept of a category is so general, it is to be expected that theorems provable for all categories will not usually be very deep. Consequently, many theorems of category theory are stated and proved for particular classes of categories. Homological algebra is concerned with Abelian categories, which exhibit features suggested by the category of Abelian groups. Logic is studied using topos theory: a topos is a category with certain properties in common with the category of sets but which allows the logic of the topos to be weaker than classical logic. It is characteristic of the malleability of category theory that toposes were originally developed to study algebraic geometry. An important use purpose of categorical reasoning is to identify within a given argument that part which is trivial and separate it from the part which is deep and proper to the particular context. For example, in the study of the theory of the GCD, the fact that it is essentially unique simply follows from the uniqueness of the product in any category and is thus really trivial. On the other hand, the fact that the GCD of the integers A and B can be expressed as a linear combination of A and B with integer coefficients—GCD(a, b) = ma + nb, for some integers m and n—is a much deeper fact that is special to a much more restricted situation.

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